# How do you use the convergence tests, determine whether the given series converges #sum (7-sin(n^2))/n^2+1# from n to infinity?

So:

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First do the limit test for convergence, and check the limit, as the limit needs to equal 0 for a series to be considered convergent, if the limit is not 0, it is then divergent.

Thus:

Therefore we can conclude that the series does not converge, and therefore is Divergent.

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To determine the convergence of the series (\sum_{n=1}^\infty \frac{7 - \sin(n^2)}{n^2 + 1}), we can use the Limit Comparison Test.

Let's denote the series as (a_n = \frac{7 - \sin(n^2)}{n^2 + 1}).

Consider a simpler series (b_n = \frac{1}{n^2}), which is a well-known convergent p-series with (p = 2).

Now, we find the limit of the ratio (L) as (n) approaches infinity:

[L = \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{7 - \sin(n^2)}{n^2 + 1}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{7 - \sin(n^2)}{n^2 + 1} \cdot n^2 = \lim_{n \to \infty} \left(7n^2 - \frac{\sin(n^2)}{n^2 + 1} \cdot n^2\right)]

Since (\lim_{n \to \infty} \frac{\sin(n^2)}{n^2} = 0), we can simplify (L) to:

[L = \lim_{n \to \infty} \left(7n^2 - \frac{\sin(n^2)}{n^2 + 1} \cdot n^2\right) = \lim_{n \to \infty} 7n^2 = \infty]

Since (L = \infty), the Limit Comparison Test implies that either both series converge or both diverge. Since (b_n) is known to converge, the original series (\sum_{n=1}^\infty \frac{7 - \sin(n^2)}{n^2 + 1}) also converges.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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